Positivity (2015) 19:289–303
A prophet inequality for L
Received: 21 March 2014 / Accepted: 21 May 2014 / Published online: 11 June 2014
© The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract Let X = (X
be a sequence of arbitrarily dependent nonnegative ran-
dom variables satisfying the boundedness condition
where t > 0, 1 < p < ∞ are ﬁxed numbers and the supremum is taken over
all ﬁnite stopping times of X.LetM = E sup
and V = sup
expected supremum and the optimal expected return of the sequence X , respectively.
We establish the prophet inequality
M ≤ V +
p − 1
and show that the bound on the right is the best possible. The proof of the inequality
rests on Burkholder’s method and exploits properties of certain special functions. The
proof of the sharpness is somewhat indirect, but we also provide an indication how
the extremal sequences can be constructed.
Keywords Prophet inequality · Optimal stopping · Bellman function ·
Research partially supported by the NCN grant DEC-2012/05/B/ST1/00412.
A. Os¸ekowski (
Department of Mathematics, Informatics and Mechanics, University of Warsaw,
Banacha 2, 02-097 Warsaw, Poland